Dr. Tobias Mai
Prof. Dr. Moritz Weber
Simon Schmidt
Operator algebras (Functional analysis II)
(Summer term 2018)News
Possible dates for oral exams are:Monday, 30 July 2018: 14:00 St.K., 14:45 NN, 15:30 NN, 16:15 NN, 17:00 NN
Tuesday, 31 July 2018: 14:00 E.W., 14:45 M.St., 15:30 L.J., 16:15 NN, 17:00 NN
Monday, 1 October 2018: 12:15 M. Sp., 14:00 A.W., 14:45 F.G., 15:30 K.K., 16:15 M.Sch., 17:00 J.Y.
Tuesday, 2 October 2018: 10:00 A.M., 10:45 J.Sch., 11:30 NN, 12:15 I.O.
In case none of the above dates suits you (for a good reason), please contact us for individual dates.
In the winter term, there will be:
- Lecture on free probability
- Reading seminar on quantum groups and Hopf algebras
- 22nd Internetseminar on Ergodic Theorems
Cohomology of quantum groups and quantum automorphism groups of finite graphs (8-12 October,
Saarbrücken). Everyone who is interested is welcome.
Lecture
Monday and Wednesday, 14-16, SR 10 (room 316, building E2 4)The language of the course is English by default, unless all participants speak German.
In this lecture, which is formally a continuation of the lecture Functional Analysis (Funktionalanalysis) held in the previous semester, we will focus on the operator algebraic aspects of functional analysis. Operator algebras are generalizations of matrix algebras to the infinite dimensional setting; they are given as subalgebras of the algebra of all bounded linear operators on some Hilbert space that are invariant under taking adjoints and closed with respect to some specific topology. Roughly speaking, operator algebras are used to study by algebraic means the analytic properties of several operators simultaneously; their theory thus combines in a fascinating way linear algebra and analysis. The most prominent examples of such operator algebras are C*-algebras and von Neumann algebras, which show a very rich structure and have various applications both in mathematics and physics, especially in quantum mechanics. Whereas the former have a more topological flavour (and their theory is thus often addressed as non-commutative topology), the latter has more measure theoretic and probabilistic sides and gives rise to non-commutative measure theory and non-commutative probability theory. We give an introduction to both the basics and some more specialized topics of the theory of C*-algebras (such as the GNS construction, their representation theory, and universal C*-algebras) and von Neumann algebras (such as factors and their classification, the hyperfinite factor, and group factors). |
Lecture Announcement
The rough plan of the lecture is the following:
- 9-11 April: Introduction/Reminder on basics in functional analysis, Outlook (Mai)
- 16 April -2 Mai: Positivity and states in C*-algebras, the GNS construction (Weber)
- 7 May - 13/20 June: Von Neumann algebras and factors (Mai)
- 18/25 June - 18 July: Examples of universal C*-algebras, group C*-algebras (Weber)
Script
Functional analysis II scriptFunctional analysis script from winter 17/18
Exercises
The exercise sessions will be held by Andreas Widenka and take place onWednesday, 16:00- 17:30 in Seminar Room 10, Building E2 4
Exercise Sheet 1
Exercise Sheet 2
Exercise Sheet 3
Exercise Sheet 4
Exercise Sheet 5
Exercise Sheet 6
Exercise Sheet 7
Exercise Sheet 8
Exercise Sheet 9
Exercise Sheet 10
Exercise Sheet 11
Exercise Sheet 12
Exercise Sheet 13 (this is the last sheet)
How to obtain the credit points
In order to obtain the credit points for this course, you must actively take part at the exercise sessions(not missing them more than twice) and obtain 50% of the total of all points on the exercise sheets.
You will then be permitted to take part at the oral exams at the end of the term which are the basis
for your grade.
References
Books:- Claire Anantharaman and Sorin Popa, An introduction to II_{1} factors, preliminary version.
- Bruce Blackadar, Operator algebras. Theory of C*-algebras and von Neumann algebras, 2006.
- Kenneth Davidson, C*-algebras by example, 1996.
- Jacques Dixmier, Les C*-algebres et leurs representations, 1969.
- Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras.
Volume I-IV, 1997. - Gerard Murphy, C*-algebras and operator theory, 1990.
- Masamichi Takesaki, Theory of Operator Algebras I-III, 2002/2003
- Script by Vaughan Jones, Berkeley, 2009.
- Lecture Notes Von Neumann algebras and ergodic theory of group actions, IHP 2011.
- Script by Jesse Petersen, Vanderbilt, 2013.
- Script by Sven Raum, Münster, 2015/2016.
- Lecture Notes Von Neumann Algebras, Subfactors, Knots and Braids, and Planar Algebras
by Roland Speicher, Saarbrücken, 2016. - Lecture Notes by Cyril Houdayer, Paris
Updated: 20 September 2018 Moritz Weber